Ismael Mejía
email: ismael.mejia@mines-nantes.fr

Mario Südholt
email: mario.sudholt@mines-nantes.fr

Towards a robust model for distributed aspects

Keywords: D.3.3 [Programming Languages]: Language Constructs and Features-Frameworks, Patterns Design, Languages Actor model, Aspect oriented programming, Distributed aspects, Distributed systems, Invasive software composition

In this paper, we present some of the problems we have found in distributed aspect models and introduce a set of criteria that we consider necessary for a robust distributed aspect system. We outline of a first version of model based on aspects and actors capable of meeting these criteria.

INTRODUCTION

Novel approaches to compose and create applications have often been developed in response to the increasing complexity of software systems. These approaches typically rely on new ways to abstract the structure of the programs with components or services which are managed by specialized software called middleware e.g., JavaEE Application servers, or the OSGi service platform.

Application developers rely on these abstractions and their corresponding tools because they help to build software applications in a more efficient way as they can concentrate on specific application needs without spending time on the details of other common requirements which are supplied by their composition model and its middleware implementation. For example, if a programmer follows the contract of the JavaEE specification when he implements a module (EJB) the application server guarantees some properties like transactionality or security. Middleware also offers tools to monitor and evolve a system during his lifetime e.g., tracing, profiling, etc.

Aspect Oriented Programing (AOP) is a complementary composition methodology introduced by Kiczales in [START_REF] Kiczales | Aspect-oriented programming[END_REF] as a partial solution to the problem of correct modularization of application concerns that crosscut many of the decomposed parts of a system. The best studied model of AOP is AspectJ [START_REF] Kiczales | An overview of AspectJ[END_REF] which is based in two mechanisms: (1) a rich declarative language which can identify the particular scattered/tangled points that are present in the application and (2) the ability to alter the current application semantics with a method-like construct called advice. AOP has been embraced immediately by system software developers as many tasks of middleware, specifically application servers fit exactly with an aspect-based adaptation model. Common scenarios where this applies are the validation of security credentials, tracing, and dependency injection.

Nowadays, most of the traditional isolated computing environments have evolved into distributed and mobile contexts. Indeed, distributed computing is now the main form of computing due to the widespread availability of the Internet and the introduction of wireless network support in mobile phones and other appliances. Distributed Aspects have been developed to augment the expressivity of the traditional aspect model with explicit distribution constructs which are able to detect join-points and execute advice on different machines in order to support the particular needs of distributed systems (e.g., coordination, concurrency, etc).

Today many middleware platforms include constructs based on AOP, such as Spring AOP, or the EJB interceptors in the case of JavaEE application servers. Nevertheless, there is not any existing middleware platform which includes distributed AOP support, even if many middleware platforms support explicit distribution requirements via message brokers, or other mechanisms. We argue that this is in part because current models for distributed aspects have been focused on the language expressivity, but not much in the the particular characteristics of the dynamic, heterogenous and asymmetric environments commonly present in distributed systems.

In this paper we study in detail some of the problems we have found through our experimentation with different distributed aspect approaches. We present and motivate why the actors models [START_REF] Hewitt | A universal modular actor formalism for artificial intelligence[END_REF], [START_REF] Agha | Actors: a model of concurrent computation in distributed systems[END_REF] seem to be a useful complement to distributed aspects. We also introduce our first approach for a design based in actors. We expect that this integration paves a way for the construction of a robust and resilient distributed aspect model that can be integrated into existing infrastructure software, and software development tools.

This paper is structured as follows. In Sec. 2 we introduce the concepts of distributed aspects, its motivation and some of the problems of adding distribution to an aspect model. In Sec. 3 we present a set of desirable characteristics for a reliable distributed aspect model, as well as our design to accomplish those characteristics inspired by the actor model. Finally in Sec. 4 we present the conclusions and some ideas for future work.

DISTRIBUTED ASPECTS: MODELS AND PROBLEMS

The evolution of distributed systems, and in particular the evolution of legacy systems is a difficult task. In this context it is common that some application concerns become distributed and hard to manage, e.g., the authentication and access rights of a user between multiple parts of a system, or the tracing of the different operations and systems involved in an electronic business transaction. Some of these tasks are commonly treated via complex refactorings of the existing applications, or via time demanding standarization processes to achieve a certain level of homogeneity between applications. Distributed aspects offer a rich and more expressive model to treat such evolution problems and enable the modularization and isolation of concerns in a nice way.

The relation between AOP and distribution was first studied as part of the work of Soares et al. [START_REF] Soares | Implementing distribution and persistence aspects with AspectJ[END_REF]. They presented a use case of the use of aspects to modularize the distribution concern of an application. This interesting use of AOP is not what we refer to as distributed aspects, but it paved a way to the analysis of the relation of aspects and distribution. Distributed aspects are an approach to have a more expressive and sophisticated aspect model to deal with the issues of the introduction of communications in software evolution.

The first model that introduced specific constructs for distributed AOP was proposed by Pawlak et al [START_REF] Pawlak | Jac: A flexible solution for aspect-oriented programming in java[END_REF]. They introduced a group of re-usable aspect components that defined distribution properties deployed in remote instances, and included broadcasting between remote objects. Nishizawa et al. [START_REF] Nishizawa | Remote pointcut: a language construct for distributed AOP[END_REF], introduced the concept of remote point-cuts to identify join points on the execution of remote hosts, but used a centralized advice server. Benavides et al. [START_REF] Luis | Explicitly distributed aop using awed[END_REF] proposed a more expressive language and model called AWED that supported not only remote pointcuts, but remote advices and asynchronous communications. Another interesting model presented by Tanter et al. [START_REF] Tanter | Expressive scoping of distributed aspects[END_REF] takes a different approach: aspects travel with the requests between different hosts and alter their semantics as they go by.

In the presence of distribution, aspects can be seen as an event-driven model: join-points are implicit events matched through a point-cut language, and advice reacts to those events via event handlers. However, a big difference between distributed event systems and distributed aspects is their goal. Event-driven systems model the flow and reaction of programs via events and handlers. Aspects care about the set of points in the system where particular events happen in order to inject new functionalities via meta-programming techniques which alter the semantics of the existing program.

A model of distributed aspects share the advantages but unfortunately the problems of both worlds. First, aspects, and in general invasive composition approaches have been criticized for being too powerful and unrestricted which can lead to errors when the semantics of the original programs is modified in unexpected ways. Second, distribution adds a new set of issues, e.g., what to do in the case of unavailable hosts, or in the case of topology changes. We present now a summary of some of the most important problems that we have found based in our work on the AWED model and invasive composition of distributed patterns::

1. Events (join-points in AO-speak) cannot be delivered to all the interested hosts (because of availability). This can lead to problems if those events should modify the behavior of other systems.

2. The correct version of the aspects should be available in order to react to the received events and broadcast event information.

3. Unrestricted access from external systems, aspects are powerful mechanisms, so it is normal that we want to have ways to control or restrict who and when can modify a system, e.g. A message from an unauthorized external entity should not be able to modify the core functionalities of a system.

4. Reactions (advices in AO-speak) cannot be processed on the designated hosts, due to (1) host unavailability or (2) different software versions, or programming errors which make those non-applicable or simply wrong.

TOWARDS AN ACTOR-BASED MODEL FOR DISTRIBUTED ASPECTS

Most of the existing research on distributed aspects focuses on the power of expression of the abstractions used to deal with the modularization of complex interactions between components and aspects in distributed settings. Nevertheless, some technical and practical consequences of the introduction of distributed aspects have not been profoundly studied. Below we present a set of characteristics that we consider important in order to achieve a robust model but that have been addressed only rudimentarily in existing systems:

• Scoping: A system with many participants and interactions needs to be controlled to be managed. One approach to achieve this is to support a model of domains for interactions between groups of participants, akin to ideas like ambients in mobile programming [START_REF] Cardelli | Mobile ambients[END_REF].

• Controlled Invasiveness: The level of invasiveness of aspects should be controllable because in many cases application programmers don't want the semantics of the core part of their programs to be modified arbitrarily and prefer to use explicit hooks to create aspectaware components (similar to the formal model of open modules [START_REF] Aldrich | Open modules: Modular reasoning about advice[END_REF]. Controlled invasion is particularly important in the presence of stringent security requirements, which are crucial for many application domains, such as large-scale service compositions.

• Mobility: Aspects should be able to move and to be applied between different parts of the system in order to improve not only its expressive power, but also its scalability. This is essential given the dynamicity of distributed systems.

• Fault-Tolerance: Communication delays and failures are the norm, not the exception, in many distributed systems and should be considered as an essential part in the design of a pragmatic model.

Our thesis is that the actor model provides a certain number of concepts and mechanisms that can be integrated advantageously with distributed aspects in order to make them more robust. In the following, we therefore briefly review the characteristics of the actor model and then introduce those of our actor-aspect model.

A brief introduction to actor models

The actor model has been introduced by Hewitt et al. [START_REF] Hewitt | A universal modular actor formalism for artificial intelligence[END_REF] as a model for concurrent computing. Actors are independent entities that encapsulate a state and a thread of control. They communicate with one another via asynchronous message passing, There are three laws that govern how an actor responds to a message: (1) they can operate at a local level, altering its current state and behavior, (2) they can send messages to other actors asynchronously, and (3) they can create other actors.

Many approaches have explored the capabilities of the actor model and refined its properties in the presence of particular conditions (e.g. migration of actors, changes of topology, guaranteed delivery of messages, reordering of messages, among others. [START_REF] Armstrong | Concurrent programming in Erlang[END_REF][START_REF] Miller | Robust composition: Towards a unified approach to access control and concurrency control[END_REF][START_REF] Dedecker | Ambient-oriented programming in ambienttalk[END_REF]).

We consider that the properties of the actor model are appropriate to fix some of the problems of the existing distributed aspect models e.g., asynchronous communication enables the decoupling of communication actions that is a big very useful at the level of distributed aspects. Another important property is the absence of shared state between actors, that provides a strict model to control the access to the actor state. This is essential to control the interaction between aspects and to a clean definition of aspect instantiation models. Locality is another useful property given that actors can only send messages to actors they know, this creates a concept of group which can be used to limit the scope of aspect application, as well as to define rules for the activation or application of collaborative aspects. Migration of actors is one of the most important properties in order to support heterogeneous and dynamic systems where actors can be moved and applied in different points. Finally concurrency is a natural characteristic of the actor model which can be exploited to reduce the overhead of the presence of aspects.

The actor model has some weak points that affect our design e.g., there are no guarantees on the order of the received messages. Another issue is that the control flow of an actor-based application often becomes difficult to follow. Finally, another issue arises in systems who need synchronous semantics as the actor model does note support it natively.

To the best of our knowledge, no existing aspect-oriented approach combines an expressive actor-like model with invasive access that supports simple and sophisticated compositions. Furthermore, we consider the presence of failure as an essential feature of our design. This is an important idea that has been elaborated only very rudimentarily in other distributed aspect models, a notable exception being work on aspect deployment [START_REF] Truyen | Run-time and atomic weaving of distributed aspects[END_REF].

Characteristics of a language and execution model for actor-based aspects

Using the common aspect language and execution dimensions introduced in [START_REF] Brichau | Survey of aspect-oriented languages and execution models[END_REF], we present some of the desired characteristics of our model derived from our design goals, and the selection of an actor-based approach.

The joinpoint model should support both static and dynamic joinpoints. Static joinpoints identify execution events, such as concrete method/function executions. Dynamic joinpoints evaluate runtime conditions which in the case of distributed aspects address not only state-related conditions, but also conditions involving locations. The pointcut language should allow the selection of events but also sequences of related events in order to respond to complex conditions involving multiple hosts. Such event sequences permit to catch situations such as a user authenticating in a given location and trying to execute a service in a different location.

Executing aspects can be mapped one-to-one to actors. In this case, we can exploit the actors' properties and need not synchronize the access to the shared mutable state: aspect instances are thus ready for concurrency, and finer instantiation models can be supported e.g., per-object, per-request and in particular per-sequence because sequences can be naturally encoded as state machines inside each actor.

Aspect composition is facilitated given the decoupled communication characteristics of actors: in most of the cases it would only be necessary to create or forward messages between actors. Deployment strategies should support dynamic weaving and take advantage of the mobility properties of the actor model.

Towards an actor-based architecture for distributed aspects

We designed an extensible architecture for aspects and actors via an aspect manager that should be installed in every host, ideally but not necessarily as part of a middleware platform, and that is going to manage execution concerns, such as communication, security, scoping, monitoring, etc.

The model of the figure 1 presents two main communication points: A component that receives incoming messages, e.g., information about events (joinpoints), new aspects to deploy, or internal communication messages, e.g., about changes in the topologies. The main role of this com-ponent (called director in the figure) is a broker that forwards messages to the corresponding parts of the system. Another important communication component is the message dispatcher. It decouples the sending of messages between nodes, to guarantee their delivery in the case of host unavailability.

The rest of the components in the figure can be classified in two groups: a group in charge of security and monitoring, and a group in charge of deploying and management of the life-cycle of aspects, including weaving and activation.

The first group is in charge of the validation of the permissions and the scope of the demanded actions e.g., messages from unrelated hosts or actions that should be ignored, or those managing which hosts have the rights to deploy aspects. It also monitors the system for anomalies and sends data to the second group. The second group controls the changes in the topology and the versions of the aspects, this implies the planning of the best strategies to be appliedwhich may include optimization ideas like batching of sequences of messages and alternative processing strategiesthe deployment and activation of aspects in the system.

Our model observes a number of interesting correctness properties, in particular, because it supports guaranteed delivery of events in the case of unavailable hosts or communication errors. It therefore addresses the issues 1 and 4 (delivery problems) presented in section 2. A control of versions is given by a combination of our security actor, and the implicit isolation properties of the actor model: our models thus covers the issues 2 and 3. The desirable technical characteristics of the model that we described previously in this section are also addressed in our model in the following way: fault tolerance thanks to the asynchronous nature of actors and its interaction properties. Manageability is addressed by means of the director and the planner components, controlled invasiveness is achieved thanks to the locality properties of the model, the deployer component, and the security manager. Finally, mobility is addressed by the actor model in a native way, but is enforced also by the security component.

CONCLUSION AND FUTURE WORK

In this paper we have detailed some problems of current distributed aspect models in the presence of common requirements of distributed systems like fault-tolerance and security. We introduced the actor model and we explained why an actor-based design is appropriate to build a more robust infrastructure for distributed aspects. We also introduced a small set of desired properties and a suitable architecture for an actor-based distributed aspect language and corresponding execution environment.

Many areas still remain to be studied in our design: several features of actors have to be explored in further detail, e.g., how to encode synchronous behavior, how to deal with intra-actor dependencies. We plan to address these issues using a more expressive integrative model and develop a first implementation of our model.

Figure 1 :

 1 Figure 1: Architecture of our actor-based framework for distributed aspects


Bruno Macke

Bernard Ségard

Comment on "Optical precursors in the singular and weak dispersion limits"

Keywords: numbers: 42.25.Bs, 42.50.Md, 41.20.Jb

We point out inconsistencies in the recent paper by Oughstun et al. on Sommerfeld and Brillouin precursors [J. Opt. Soc. Am. B 27

In a recent paper [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], Oughstun et al. revisit the classical problem of the propagation of a step modulated pulse in a Lorentz model medium. They specifically consider the case where the absorption line is narrow (singular limit) and the one where the refractive index of the medium keeps very close to unity at every frequency (weak dispersion limit). The medium is characterized by its complex refractive index

n(ω) = 1 - ω 2 p ω 2 -ω 2 0 + 2iδω 1/2 . (1)
Here ω, ω p , ω 0 and δ respectively designate the current optical frequency, the plasma frequency, the resonance frequency and the damping or relaxation rate. The wave propagates in the z-direction. In the following we use the retarded time t, equal to the real time minus z/c where c is the velocity of light in vacuum (retarded time picture). The medium is then characterized by the transfer function

H(ω) = exp i ω c z (n(ω) -1) (2)
and the field transmitted E(z, t) at the abscissa z reads as

E(z, t) = 1 2π ˆ+∞+ia -∞+ia H(ω) E(0, ω) exp(-iωt)dω. ( 3
)
Here a is a positive constant and E(0, ω) is the Fourier transform of the incident field E(0, t). In [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], the latter is assumed to have the idealized form

E(0, t) = Θ(t) sin(ω c t) (4)
where Θ(t) is the unit step function and ω c is the frequency of the optical carrier.

Although it abundantly refers to the theoretical results obtained by the asymptotic method, the study reported in [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF] is mainly numerical. All the simulations are made for ω 0 = 3.9 × 10 14 rad/s [2] and ω c = 3.0 × 10 14 rad/s in a normal dispersion region. The singular and weak dispersion limits are respectively attained when δ ≪ ω 0 and ω 2 p ≪ δω 0 [see Eq. ( 1)]. Oughstun et al. emphasize that these two limiting cases "are fundamentally different in their effects upon propagation" but, surprisingly enough, they take for their simulations in the weak dispersion limit a value of δ for which the singular limit nearly holds (δ < ω 0 /100). Consequently, mutatis mutandis, the results obtained in the two limits appears qualitatively similar. The steep rise of the Brillouin precursor obtained in the singular limit (their Fig. 3) and analyzed as a distinguishing feature of this limit is only due to an unsuitable time scale. As shown in our Fig. 1, obtained for the same values of the parameters, the rise is quite progressive and well reproduced by a Airy function. This result, established by Brillouin himself in 1932 [START_REF] Brillouin | Propagation des ondes électromagnétiques dans les milieux matériels[END_REF], is easily retrieved from Eqs (1), (2), and (3). Anticipating that the beginning of the precursor involves frequencies ω such that δ ≪ ω ≪ ω c , we use the approximate relations

n(ω) ≈ 1 + ω 2 p ω 2 0 1/2 + ω 2 ω 2 p 2ω 3 0 ω 2 0 + ω 2 p 1/2
(5) and E(0, ω) ≈ 1/ω c . Introducing the new retarded time [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], that is for ω0 = 3.9 × 10 14 rad/s, ωc = 3.0 × 10 14 rad/s, ωp = 3.05 × 10 14 rad/s , δ = 3.02 × 10 10 rad/s, and z = 7.232 × 10 -2 m. The full line is the exact numerical solution obtained by fast Fourier transform (FFT) and the dashed line is the approximate analytical solution given by Eq.7.

t ′ = t -t b with t b = z c 1 + ω 2 p /ω 2 0 1/2 -1 ,
We finally get

E(z, t ′ ) = 1 2πω c ˆ+∞ -∞ exp iω 3 3b 3 -iωt ′ dω = b ω c Ai(-bt ′ )
(7) where Ai(x) is the Airy function [START_REF]3 x) . Peak amplitudes of A(x) and Ai[END_REF]. As it appears Fig. 1, this analytic expression perfectly fits not only the rise of the precursor but also a significant number of its subsequent oscillations. Equations ( 6) and ( 7) also provide a simple evidence of the z -1/3 dependence of the precursor amplitude on the propagation distance. Figure 2 shows the Brillouin precursor obtained in the conditions of the Figure 5 of [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], intended to illustrate the specific case of the weak dispersion limit. As indicated previously, the conditions of the singular limit are approximately met and this explains why the first oscillation of the precursor and thus its peak amplitude are well reproduced by Eq. (7). Note that the rise-time of the precursor, proportional to 1/b, is 6 times faster than in the previous case.

In their study of the weak dispersion limit, Oughstun et al. [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF] mention a "curious difficulty in the numerical FFT simulation of pulse propagation" and, in order to overcome it, they develop at great length an equivalence relation. In fact the difficulty is completely avoided in the retarded-time picture and their equivalence relation then appears as an immediate result. In the weak dispersion limit, the transfer function, as given by Eq.(2), is indeed reduced to

H(ω) ≈ exp - iω 2c ω 2 p z ω 2 -ω 2 0 + 2iδω . ( 8
)
This only expression shows that, for given ω 0 and δ, all 1 for the parameters of the figure 5 of [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], that is for ω0 = 3.9 × 10 14 rad/s, ωc = 3.0 × 10 14 rad/s, ωp = 3.05×10 12 rad/s , δ = 3.02×10 12 rad/s, and z = 2.290m. The rapid oscillations of small amplitude superposed upon the beginning of the Brillouin precursor are the end of the Sommerfeld precursor.

the media having the same ω 2 p z (and thus the same optical thickness at any reference frequency) are equivalent.

In a real experiment, the incident field is obviously turned on in a finite time. Oughstun et al. state that their "results will remain valid for a non instantaneous turn-on signal provided that the signal turn-on time T r is faster than the characteristic relaxation time 1/δ". This is grossly false. Figure 3 shows the results of a simulation performed for the parameters of their figure 3 and 10 -90% turn-on times (a) T r = 0, (b) T r = τ /500, and (c) T r = τ /100 where τ = 1/δ is the relaxation or damping time (τ ≈ 33ps). The rise of the incident field is modelled by replacing the step-function Θ(t) appearing in the idealized form of Eq. ( 4) by f (t) = 1 2 [1 + erf(ξt)] where erf(x) designates the error function. The corresponding rise-time is T r ≈ 1.8/ξ and f (t) → Θ(t) when ξ → ∞. We see that the effect of the rise-time is actually catastrophic. For a rise-time as fast as τ /500 (≈ 66fs ), the Sommerfeld precursor is absent. The maximum of the Brillouin precursor is significantly time-delayed and its amplitude A B is reduced by a factor of about 300, becoming comparable to the amplitude e -10 of the "main field". The reduction factor obviously depends on the optical thickness and, consequently, the power law A B ∝ z -1/3 , obtained for T r = 0, breaks down. Finally the curve (c) of Fig. 3 shows that both precursors practically vanish for T r = τ /100 . By means of other simulations, we find that the Sommerfeld precursor is very attenuated as soon as T r > τ /5000 ≈ 2/ω c and that a correct reproduction of both precursors as obtained for T r = 0 requires that T r ≤ 1/ω c . This condition results from the fact that the Sommerfeld (Brillouin) precursor mainly involves frequencies ω ≫ ω c (ω ≪ ω c ) and thus is excited by the corresponding frequencies contained in the spectrum of the incident field. From the expression of

(c)

Figure 3: Effect of the rise-time of the incident field on the transmitted field. The parameters are those of the figure 3 of [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], that is ω0 = 3.9 × 10 14 rad/s, ωc = 3.0 × 10 14 rad/s, ωp = 3.05 × 10 14 rad/s , δ = 3.02 × 10 10 rad/s, and z = 7.232 × 10 -2 m. The vertical dashed line indicates the group delay τg = dϕ dω |ω=ω c where ϕ is the argument of H(ω) (τg = 471ps). τg fixes the arrival of the "main field" whose amplitude, equal to exp (-10) ≈ 4.5 × 10 -5 , obviously does not depend on the rise-time. The different curves are obtained for (a) Tr = 0, (b) Tr = τ /500, and (c) Tr = τ /100 where τ = 1/δ is the relaxation or damping time of the medium. f (t), it is easily shown that the effect of a finite rise-time T r (T r ≈ 1.8/ξ ) is to divide E(0, ω) by exp ω 2 / 4ξ 2 when ω/ω c → ∞ and by exp ω 2 c / 4ξ 2 when ω/ω c → 0. These expressions explain why the Sommerfeld precursor is much more affected by the rise-time effects than the Brillouin precursor and enable one to predict that, for T r = 1/ω c (ξ ≈ 1.8ω c ), the amplitude A B of the Brillouin precursor will be about 7.5% below that obtained for T r = 0 (result confirmed by an exact numerical calculation).

The observation of a signal close to that shown on the figure 3 of [START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF] requires not only that the rise-time of the incident field does not exceed 1/ω c (3.3fs) but also that its subsequent amplitude remains nearly constant during a time that would be more than four orders of magnitude longer. The fulfilment of this double condition, either with a pulsed laser or with a continuous wave laser followed by a modulator, appears to be quite unrealistic from an experimental viewpoint. More generally, due to such temporal constraints, we don't see how the Brillouin precursor could be actually used in optics as a tool for imaging through a dense medium, opaque at the carrier frequency. Anyway, the problem of optical precursors is of fundamental interest from a theoretical viewpoint, even if the experiment imagined by Sommerfeld and Brillouin may appear as a gedankenexperiment. In this spirit, we are examining what the precursors become when the incident field E(0, t) = Θ(t) sin(ω c t), constantly considered in the theoretical papers, is replaced by E(0, t) = Θ(t) cos(ω c t). The true discontinuity of the incident field at the initial time then leads to radically new effects that we will discuss in a forthcoming paper.

the transfer function then reads as exp iω 3 / 3b 3 where b = ω 0 2c ω 2 0 + ω 2 p 1 Figure 1 :

 311 Figure1: Brillouin precursor obtained for the parameters of the figure3of[START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], that is for ω0 = 3.9 × 10 14 rad/s, ωc = 3.0 × 10 14 rad/s, ωp = 3.05 × 10 14 rad/s , δ = 3.02 × 10 10 rad/s, and z = 7.232 × 10 -2 m. The full line is the exact numerical solution obtained by fast Fourier transform (FFT) and the dashed line is the approximate analytical solution given by Eq.7.

Figure 2 :

 2 Figure2: Same as Fig.1for the parameters of the figure5of[START_REF] Ke Oughstun | Optical precursor in the singular and weak dispersion limits[END_REF], that is for ω0 = 3.9 × 10 14 rad/s, ωc = 3.0 × 10 14 rad/s, ωp = 3.05×10 12 rad/s , δ = 3.02×10 12 rad/s, and z = 2.290m. The rapid oscillations of small amplitude superposed upon the beginning of the Brillouin precursor are the end of the Sommerfeld precursor.


Bruno Macke

Bernard Ségard
email: bernard.segard@univ-lille-1.fr

Optical precursors in the singular and weak dispersion limits: Reply to reply to comment

Keywords: numbers: 42.25.Bs, 42.50.Md, 41.20.Jb

In a recent comment [1], we pointed out inconsistencies in the paper by Oughstun et al. on Sommerfeld and Brillouin precursors [2]. Their reply [START_REF] Oughstun | Optical precursor in the singular and weak dispersion limits: Reply to Comment[END_REF] makes no response to the main points raised in our comment. In addition it is written in such a manner that it may confuse and mislead the unfortunate reader. It thus appears necessary to dot the i's and cross the t's.

The first point concerns the analytical form of the Brillouin precursor. No need to look at the Oughstun's papers abundantly cited in [2, 3] for (possibly) finding the remarkably simple expression E B (z, t ′ ) = b ωc Ai(-bt ′ ). As mentioned in our comment [1], this expression was established by Brillouin himself as far back as 1932 [START_REF] Brillouin | Propagation des ondes électromagnétiques dans les milieux matériels[END_REF]. The way by which we have retrieved this result shows that it will be a good approximation of the exact signal when the instantaneous frequency of the precursor is large (small) compared to the damping rate δ (the carrier frequency ω c ). This condition is met in the singular dispersion limit, at least for the first oscillations of the precursor. Since b ∝ z -1/3 (z propagation distance), the previous result analytically shows that the amplitude of the Brillouin precursor also scales as z -1/3 , a result that was obtained only numerically in [2]. The expression of E B (z, t ′ ) also shows that the instantaneous period of the precursor is proportional to 1/b. In particular, the first half-period HP 1 of the precursor, defined as the time interval between the first maximum and the first minimum of E B (z, t ′ ), is equal to 2.22/b. For the parameters used to obtain Fig. 3 in [2], we get HP 1 = 0.23 ps in agreement with the exact result (see Fig. 1

of our comment [1]

). On their side, Oughstun et al. give in the equation (15) of [2] an estimate of the half-period that, in the same conditions, leads to a value smaller than the right one by about five orders of magnitude! Contrary to the claim made in [2], we have shown in our comment [1] that the precursors are catastrophically affected by the rise-time T r of the incident field even when the latter is considerably faster than the damping time τ = 1/δ of the medium. Our analysis was supported by numerical simulations made for rise times as fast as τ /500 and τ /100. Without disputing the validity of our simulations, Oughstun et al. claim in their reply that our numerical results "do not properly lie in the asymptotic regime of sufficiently large propagation distances" [START_REF] Oughstun | Optical precursor in the singular and weak dispersion limits: Reply to Comment[END_REF]. In fact, the cases considered in our simulations are exactly those considered in their paper [2]. In particular, as noticed both in the text and in the figure caption, the parameters of our Fig. 3 are strictly identical to those of their Fig. 3. However, as suggested by Oughstun et al., we have made additional simulations for larger propagation distances. These simulations show that the risetime effects become more and more catastrophic when z increases. This result was obviously foreseeable. Indeed the effect of the rise-time, irrespective of the precise shape of the rise, is to contract the spectrum of the incident field about the carrier frequency ω c . On the other hand, when z increases, the medium becomes strongly absorbing on a broader and broader spectral region, keeping some transparency only at frequencies ω ≫ ω c and ω ≪ ω c . The precursors, associated with the transparency regions, then cannot be excited by the incident field.

Oughstun et al. also invoke in their reply [START_REF] Oughstun | Optical precursor in the singular and weak dispersion limits: Reply to Comment[END_REF] an eventual interaction between the Brillouin precursor and the so-called main signal. Such an interaction, if it exists, is extremely small in the conditions of our figure 3 for which the amplitude A MS of the main signal is only e -10 , that is about 3700 times smaller than the amplitude A B of the Brillouin precursor. The interaction becomes completely negligible when the propagation distance is increased in order to better "lie in the asymptotic regime". When the optical thickness at ω c is equal to 100 (A MS = e -100 ≈ 3.7×10 -44 ), we have determined the amplitude of the Brillouin precursor for T r = 0 and T r = τ /500 (error-function envelope signal) in the conditions of Fig. 3 and Fig. 5 in [2]. These figures were intended to respectively illustrate the singular and weak dispersion limits. We find A B (0) ≈ 7.9 × 10 -3 and A B (τ /500) ≈ 6.7 × 10 -10 in the former case; A B (0) ≈ 3.6 × 10 -2 and A B (τ /500) ≈ 2.2 × 10 -8 in the latter one. These results confirm in both cases the catastrophic effect of the rise time of the incident field even when it is 500 times smaller than the damping time τ = 1/δ of the medium.

In summary, the analysis presented in our comment [1] on the paper [2] is entirely confirmed.


Gonéri Le Cozannet

Anny Cazenave

David Salas Y Mélia

Guy Wöppelmann

Vincent Donato

Patrice Walker

Philippe Rogel

Manuel Garcin

Marissa L Yates

Sea level rise implications on shoreline changes: expectations from a retrospective analysis

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Sea level rise implications on shoreline changes: expectations from a retrospective analysis

Gonéri Le Cozannet 1 , Anny Cazenave 2 , David Salas Y Mélia 3 , Guy Wöppelmann 4 , Vincent Donato 5 , Patrice Walker 6 , Philippe Rogel 7 , Manuel Garcin 1 , Marissa Yates Sea level rise is a major consequence of climate change and is expected to threaten many lowlying and highly populated areas in the world. However, our understanding of the potential consequences of this phenomenon is limited by (1) the availability of validated sea level time series at regional scales, and more importantly, (2) a relatively poor understanding of the causes of long term coastal geomorphological changes. In some regions, the rate of sea level rise is up to 3 times the global mean rate (estimated as 3.5 mm per year since the early 1990s); in other regions, the rate is slower than the global mean and sea level may even be falling. The main factor causing this regional variability is the non-uniform warming of the oceans, but other phenomena such as post-glacial rebound and gravitational effects associated with land ice melt also give rise to regional variability in absolute sea level change. Thus, in the past 50 years, sea level has not risen homogeneously on the global scale, which provides an opportunity to assess changes in regions where sea level increased (or decreased) the most. This paper presents the overall approach of the CECILE project, whose objective is to assess the impacts of sea level variations on shoreline change. It focuses on assessing coastal impacts caused by sea level rise: firstly, the issue of the availability of regional sea level variation time series is addressed by merging the expertise of coastal geomorphologists with sea level experts (including integrating observations, geodesy, tide-gauge time series analysis, and climate modelling). For each site of interest, this includes the identification and quantification of all components of absolute and relative mean sea level variations, such as climate-induced sea level variations and continental subsidence and uplift at various spatial scales. Secondly, because of the practical limitations of long term coastal modelling, improving our understanding of long term modelling is approached using historical observations, in which we analyse shoreline changes at a few number of tropical sites instead of focusing in a single site. This can be done using airborne remote sensing photographs acquired in the 50's, thus providing an integrated spotlight of coastal geomorphological changes in these areas (see Garcin et al. in this session. This approach pays particular attention to assessing changes that can be attributed to humans, as well as those that occur after specific extreme events. The knowledge database acquired through this process is expected to serve as an analogue for experts trying to assess the potential future impact of sea level rise, as well as to support the creation of a coastal geomorphological classification based on shoreline vulnerability to sea level rise and climate change.

This communication is supported by the French National Agency for Research (ANR) within its Planetary Environmental Changes (CEP) framework. We thank the French Polynesia Government (SAU) and the New Caledonia Government (DIMENC, DITTT) for providing data.


Alexis Descatha
email: alexisdescatha<adale@dom.wustl.edu

Ann-Marie Dale

Alfred Franzblau

Justin Coomes

Bradley Evanoff

Comparison of research case definitions for carpal tunnel syndrome

Keywords: median neuropathy, population study, nerve conduction studies, screening METHODS carpal tunnel syndrome AND case definition, consensus definition, diagnostic criteria, OR case criteria

Objective

The aim of our study was to assess agreement between different case definitions of carpal tunnel syndrome (CTS) for epidemiological studies.

Methods

We performed a literature search for papers suggesting case definitions for use in epidemiological studies of CTS. Using data elements based on symptom questionnaires, hand diagrams, physical examinations and nerve conduction studies collected from 1107 newly-hired workers, each subject in the study was classified according to each of the case definitions selected from the literature. We compared each case definition to every other case definition, using the Kappa statistic to measure pairwise agreement on whether each subject met the case definition.

Results

We

found six unique papers in a twenty year period suggesting a case definition of CTS for use in population-based studies. We extracted seven case definitions. Definitions included different parameters: symptoms only, symptoms and physical examination, symptoms and either physical examination or median nerve conduction study, symptoms and nerve conduction study. When applied to our study population, the prevalence of CTS using different case definitions ranged from 2.5 to 11.0 . The percentage of % % misclassification was between 1 to 10 , with generally acceptable levels of agreement (Kappa values ranged from 0.30 to 0.85). % Conclusion Different case definitions resulted in widely varying prevalences of CTS. Agreement between case definitions was generally good, particularly between those that required very specific symptoms or the combination of symptoms and physical examination or nerve conduction. The agreement observed between different case definitions suggests that the results can be compared across different research studies of risk factors for CTS.

INTRODUCTION

Carpal Tunnel Syndrome (CTS) is a common and costly disease among working-aged adults ( ). The prevalence ranges from 1 [START_REF] Van Rijn | Associations between work-related factors and the carpal tunnel syndrome--a systematic review[END_REF] 1

%-%
among the general population, and up to 14.5 among specific occupational groups ( , ). Many studies of CTS have examined potential

% 1 2
risk factors, preventive measures, and interventions. However, there is no gold standard for CTS diagnosis, nor is there consensus on the " " most appropriate research case definitions for CTS. Although case definitions in published studies have used some combination of symptoms, nerve conduction testing, and/or physical exam measures, they do not agree on what methods should be used, nor on specific criteria or cut-points for testing ( ). A recent systematic review of classification and case definitions of work-related upper extremity 3 disorders retrieved seven different case definitions of CTS ( ). In a recent review that examined 44 papers dealing with the potential 4 association between occupational exposure and CTS, a large variety of case definitions were described, and only 19 of these studies used a case definition that required both typical symptoms and electrodiagnostic examination ( ). 5

Although there have been studies examining the sensitivity and specificity of individual case definitions ( , ), or some element of 6 7 these definitions ( ), these studies have not always been carried out in a general population or workplace settings, and the assessment 8 -10 of overlap between different case definitions has been limited. As noted in a 1998 consensus case definition of CTS, the proliferation of case definitions can make it difficult to compare results across studies, and assessment of agreement of case definitions may facilitate comparisons between studies that employed different case definitions ( ).

The objective of our study was to compare agreement between different case definitions of CTS for epidemiological research or population studies. We applied different case definitions of CTS to subjects in a large population of newly-hired adults and explored differences of case classification. literature search included the 20 years between 1989 to early 2009, and was limited to English language and Human subjects. Identified papers were read by two independent reviewers (AD and BE). Inclusion required consensus that a paper clearly proposed a case definition of CTS for use in epidemiological studies.

Quantitative analyses of the criteria used

Subject Recruitment and Eligibility

To assess the concordance of results from the various case definitions,, we used data from the Predicting Carpal Tunnel Syndrome

Data Collection

Subjects were tested at the time of enrollment in the study. Testing consisted of a self-administered questionnaire, a physical examination of the upper extremities, and nerve conduction studies of both hands. All examiners were members of the research team that included an occupational physician, three occupational therapists, a physical therapy assistant, an occupational therapy assistant, and three medical students. Each examiner was instructed in a standardized physical examination testing procedure and demonstrated proficiency before collecting study data. Periodic re-evaluation of examiners performance was assessed over the course of the study.

'

Symptom definition

Symptoms of the hand and wrist were assessed with a self-administered questionnaire, using the following initial question: In the past was resolved by consensus ( ). 14

Physical examination testing

The physical exam included inspection, Semmes-Weinstein sensory testing (the examiner tested light touch sensation using a monofilament applied to the distal phalanx of the long finger of each hand. An abnormal response was the inability to detect touch with a # 2.83 monofilament at least two out of three times), Tinel s test (the examiner tapped firmly over the median nerve from the palm of the ' hand to the proximal wrist. An abnormal response was recorded if the subject reported symptoms of paresthesia, burning, or numbness in the median nerve distribution), and Phalen s maneuver (the subject held the wrists in full flexion for one minute by placing the backs of the ' hands together with the elbows raised to shoulder height. An abnormal response was recorded if the subject reported symptoms of paresthesia, burning, or numbness in the median nerve distribution).

Nerve conduction testing

Examiners performed median and ulnar nerve conduction studies at the wrist bilaterally using the NC-Stat nerve conduction testing device (NEUROMetrix, Inc., Waltham, MA). This clinical tool has been found to have reliability and criterion validity similar to traditional methods of nerve conduction testing ( , ). Prior to data collection, all examiners demonstrated proficiency in use of the 15 16 device following the standard testing procedures recommended by the manufacturer. The NC-Stat required placement of self-adhesive electrodes at the wrist and fingers using anatomic landmarks; the distance in centimeters between the wrist crease and the finger electrodes was measured as part of the testing protocol. We then measured median and ulnar distal motor latencies (wrist-thenar eminence and wrist-hypothenar eminence) and distal sensory latencies (wrist-third finger and wrist-fifth finger). Because the NC-Stat sensory electrodes are placed by reference to anatomic landmarks (the distal wrist crease and the finger crease of the proximal interphalangeal joint), the distance between the wrist and finger electrodes for median nerve measurements varied between 10.2 and 17.4 centimeters in our subjects.

We normalized the measured sensory latencies for each subject to standard 14 centimeter sensory latencies using the measured nerve conduction velocity. We calculated median-ulnar sensory latency difference (MUDS), based on the 14 cm-adjusted sensory latencies.

Abnormal median nerve conduction was defined in our study as a 14 cm sensory latency of the median nerve >3.5 ms OR motor latency of the median nerve >4.5 ms OR paired transcarpal sensory difference (between median and ulnar nerves, MUD) of >0.5 ms.

Analyses

To compare various definitions, we attempted to map all elements of each identified case definition to all subjects in our study. Each subject was classified separately for each case definition using the data elements collected in the study (i.e., symptoms, hand diagrams, physical exams, and nerve conduction results). A subject was counted as a case of carpal tunnel syndrome for a particular case definition

" "
if the entire criteria were met within each arm, for either the right or the left hand. For instance, a subject with symptoms in the right hand but abnormal median nerve conduction values in left hand was not considered a case ; a subject whose symptoms and abnormal median " "

nerve conduction findings both occurred in the left hand was considered a case . We calculated the prevalence of CTS using each case

" "
definition and compared concordance between the different definitions. We compared each case definition against every other definition as the reference , and assessed intermethod agreement using the Kappa statistic to measure pairwise agreement on whether each subject met

" "
the case definition. ( , ). Values of kappa greater than 0.75 are considered excellent, values between 0.40 and 0.75 are fair to good, 17 18 and values of less than 0.40 represent poor agreement beyond chance alone ( ). Because the prevalence of CTS was low, the Kappa 19 statistic may be low despite high agreement ( ). To assess the paradox of low Kappa values despite high agreement, we also report the 20

" "
true positive, false positive, true negative, and false negative rates, and the frequency of cases misclassified ( ).

RESULTS

Literature review

The literature review produced 324 papers based on the selection criteria and cross-references. Of these, 318 did not propose case definition criteria for CTS in epidemiological studies, or cited another reference already selected. We found six papers proposing case definitions for population-based studies ( ) ( , , , ). One paper proposed 16 levels of case definition; we chose the most Table 1 7 11 13 22 -24 inclusive definition and the definition that had the greatest accuracy when compared to electrodiagnostic studies (best compromise

" "
between sensitivity and specificity) ( ). 7

All of the definitions were different: two papers suggested a definition based on symptoms only, one on symptoms plus physical examination only, two required combinations of symptoms plus either physical exam or nerve conduction studies, and one required symptoms and nerve conduction studies. The symptoms criteria varied across definitions from non-specific hand symptoms (numbness, tingling, burning or pain in the hand, and nocturnal symptoms) to specific hand symptoms as described by the Katz hand diagram ( ). 13

Quantitative analyses of research case definitions

The cohort included 435 apprentice construction workers, 478 hospital workers, 158 workers in computer or laboratory jobs, and 37 in other positions. There was wide variability in prior jobs reported, with 258 job titles ( ). The study group was 65.1 male, with a mean 12 % age of 30.8 years (SD 10.3) and a mean body mass index of 28.5 (SD 6.6). Ten subjects (1.0 ) had missing data in the symptoms, physical % examination or nerve conduction studies, and were excluded from analyses.

A list of each of the definitions we selected, and how we mapped their criteria to our methods is found in . Most of the Table 1 parameters were easily mapped using data collected in our study. Exceptions included motor weakness of the abductor pollicis on physical examination, which was not tested in our study. Instead of motor weakness, we used muscle wasting on inspection (not found for any subject in our study). We tested sensory deficits using Semmes-Weinstein monofilaments instead of two-point discrimination as required by some criteria. Our symptom criteria required recurrent or persistent symptoms in the past year, which we substituted for the temporal requirement of symptoms in the Sluiter case definition ( Symptoms present now or present on at least 4 days during the last 7 days ).

" "

Previous results have shown that averaging of symptom reporting over some period of time (e.g., symptoms in the last 7 days, 30 days, or

' '
the last year) produces relatively stable results, and so our modification of the Sluiter case definition likely had only a small impact ( ). 25

The prevalence of carpal tunnel syndrome varied from 2.5 to 11.0 , depending on the case definition used ( ). The case % Table 2 definitions requiring symptoms alone, in any part of the hand or fingers (Franzblau ( ), and Franzblau ( )) resulted in the highest 1 2 numbers of cases ( ). The case definition requiring symptoms specific to the median nerve distribution plus electrodiagnostic Table 3 abnormality (Rempel ( )) had the lowest number of cases, while those requiring symptoms plus physical exam or electrodiagnostic 11 abnormality were intermediate. When each case definition was tested against all other case definitions, we found relatively small percentages of misclassification (1 to 10 , ). The concordance using the Cohen s Kappa statistic ranged from 0.30 to 0.81. The % Table 4 ' greatest degree of misclassification (>4.3 ) and the lowest agreement measured by Kappa (<0.5) was seen when the least restrictive case % definitions -those requiring only non-specific hand or finger symptoms -were compared against case definitions requiring more specific hand symptoms (Katz hand diagram) and case definitions requiring symptoms plus physical examination or electrodiagnostic abnormality (

). Use of a Katz hand diagram alone showed good to excellent agreement with case definitions requiring specific hand figure 1 " "

symptoms and physical examination or electrodiagnostic abnormality (Kappa 0.64 0.80) when a Katz reading of probable or classic definition was required. Similar agreement was seen between case definitions requiring symptoms and physical examination or electrodiagnostic abnormality (Kappa 0.70 0.81). Slightly lower agreement was seen between a case definition that required symptoms and electrodiagnostic abnormality and those that allowed symptoms and physical examination (Kappa 0.53 0.68).

-

DISCUSSION

Different case definitions for carpal tunnel syndrome have been used in epidemiological studies, based on different combinations of symptoms, physical examination, and nerve conduction studies. When we tested different case definitions in the same study population, we found widely varying estimates of prevalence, yet a relatively high degree of quantitative concordance between case definitions, with relatively low rates of misclassification.

Not surprisingly definitions based on non-specific hand symptoms only led to the highest prevalence of disease, while more restrictive definitions requiring specific hand symptoms plus median nerve conduction abnormalities resulted in the lowest prevalence. Results of population surveillance studies are clearly sensitive to the case definition ( , ). The proportion of misclassification between more 2 26

restrictive and less restrictive case definition of CTS was relatively low.

The selection of the case definition papers was based on a literature search. We decided to include only those papers suggesting a case definition for use by other investigators, rather than testing the much larger group of different case definitions used in epidemiological studies ( , , ). We did not study definitions based on insurance or medical treatment claims of carpal tunnel syndrome nor self-reports 3 5 27 of treatment or diagnosis, though such case definitions are useful for some surveillance and epidemiological studies ( ). We believe 28 -31 that our choice of case definitions is representative of the spectrum of definitions that have been used.

One limitation of our study approach could be in the mapping of our study data to the case definitions described in the literature. When authors described a symptom or a sign not recorded in the study, we selected the closest item in our study. Only a few items were different: motor loss in physical examination, choice of sensory examination, and the time period for symptoms in the Sluiter et al. case definition.

The Sluiter time period, that incorporates a timeframe, frequency and duration, were included in their document in order to differentiate common aches and pain from work related musculoskeletal disorders, and did not serve to define the type of the disease ( ). As noted 24 earlier, the difference in time frames between Sluiter and our study probably made only minor differences in the prevalence of CTS. Motor loss in our study was evaluated by inspection (thenar atrophy). The study was based on screening a large population for clinically unreported CTS and no atrophy was found, which is not surprising. In a study of active workers, the prevalence of motor loss corresponding to severe CTS is expected to be low, even if assessed by physical examination of motor strength. Different results might be seen in a clinical population with a higher prevalence and greater severity of CTS ( ). 32

Another potential limitation of our study was our definition of abnormal median nerve conduction. In both clinical settings and population studies, the determination of normative values for nerve conduction is complicated, with different possible cut-offs depending on the studies purposes ( , , ). None of the case definitions selected from the literature defined cut-points for nerve conduction. We ' 11 33 34 chose nerve conduction cut-offs that have been proposed for use in a large multi-center study of CTS in working populations, and applied these same criteria to all case definitions. More stringent criteria for abnormality may have resulted in slightly different study results, with fewer subjects rated as abnormal ( ). 33

Our study compared concordance between different case definitions, but did not propose a best case definition for CTS -even in others. Their use depends on the purpose of the study and their feasibility ( ). We found a fair degree of agreement between different 43 case definitions in a general working population. These results suggest that comparison of risk factors for CTS across studies may not be greatly biased by misclassification errors due to differences in case definitions of CTS, though it is important to note that the prevalence of disease is likely to be different when using different case definitions. Symptoms suggestive of carpal tunnel syndrome are present. One or more of the following symptoms are sufficient: paresthesia, hypoesthesia, pain or numbness affecting at least part of the median nerve distribution of the hand(s). The median nerve distribution generally includes palmar side of thumb, index finger, middle finger, and radial half of ring finger; dorsal (back) side of same digits above PIP (proximal interphalangeal) joint; and radial half of palm. Pain and paresthesia may radiate proximally into the arm. Symptoms should have lasted at least one week or, if intermittent, have occurred on multiple occasions. Other causes of hand numbness or paresthesia, such as cervical radiculopathy, thoracic outlet syndrome, and pronator teres syndrome, should be excluded by appropriate clinical evaluation. * AND Objective findings consistent with carpal tunnel syndrome are present in the affected hand(s) and wrist(s): EITHER I)

Physical examination findings.

One or more of the following findings should be present: 1) Tinel s sign (paresthesia elicited or accentuated by gentle percussion over the carpal ' tunnel), 2) present or positive Phalen s test (paresthesias are elicited or accentuated by maximal passive flexion of the wrist for one minute), or 3) ' decreased or absent sensation to pin prick in the median nerve distribution of the hand.

Figure 1

 1 Figure 1 Distribution and median value of kappa values showing agreement between each case definition compared to all others tested. * (N 1107, = number of carpal tunnel syndrome cases mentioned below of each case definition) The top bar is the 95th percentile, the top of the box is the * 75th percentile, the horizontal line in the box is the median, the bottom of the box is the 25th percentile, and the bottom bar is the 5th percentile

OR 2 ) 13 Katz

 213 Electrodiagnostic findings of median nerve dysfunction across the carpal tunnel. Criteria for abnormal electrodiagnostic findings are generally determined by the individual laboratories. median innervated fingers (digits 1, 2, or 3) is shaded regardless of shading in palm and back of hand ie hand diagram coded as possible, classic or probable * AND (Tinel or Phalen signs, sensory loss by using Semmes Weinsteins test) Hand diagram: classic or probable Tingling, numbness, burning, or decreased sensation with or without pain in at least 2 of the digits 1, 2, or 3. = Palmar symptoms allowed if confined solely to the ulnar apect; 5th finger symptoms, wrist pain or radiation proximal to the wrist allowed. If dorsal symptoms are present, classification is possible.

Table 1

 1 Selected papers and case definition, with the mapping for the PrediCTS study

	Authors
	(first)	Original criteria

Table 2

 2 Phalen s test positive, nocturnal exacerbation of symptoms, motor loss with wasting of abductor pollicis brevis, and abnormalThe combination of electrodiagnostic study findings and symptom characteristics provides the most accurate carpal tunnel syndrome diagnosis. No single best scheme has emerged for assessing symptom qualities. A recommended classification scheme for symptoms based on Katz diagram (requires documentation of symptom location and character (numbness, tingling, burning, or pain). Electrodiagnostic studies should be performed according to the current and future guidelines prepared by the American Academy of Neurology, the American Association of Electrodiagnostic Medicine.Flexion or carpal compressions test, Tinel s or Phalen s tests, two-point discrimination, or resisted thumb abduction or motor loss with wasting of Frequency and proportion of clinical items (symptoms or physical examination), nerve conduction abnormality, using the selected case definitions in the PrediCTS study (n 1097)

						Symptoms only Katz hand diagram: classic or
						probable	(ie	Tingling,
				"	"	numbness, burning or pain in
						at least 2 of the digits 1, 2, or
						3.
						Symptoms in dorsum of hand
						excluded; wrist pain or
						radiation proximal to the wrist
						allowed; palmar symptoms
						allowed
	Franzblau-1	Numbness, tingling or pain in the hands or any finger	Symptoms only Numbness, tingling, burning
	et al, 1993	(			or pain in the hands or any
	7	)				finger
	Franzblau-2	Numbness, tingling or pain in the hands or any finger) AND (nocturnal symptoms)	Symptoms only Numbness, tingling, burning
	et al, 1993	(			or pain in the hands or any
	7	)				finger symptoms	AND	nocturnal
	Harrington Pain, or paraesthesia, or sensory loss in the median nerve distribution and one of:	Symptoms

] • * In the validation study of the Matte et al definition, the Katz hand diagram was used as the criteria for symptoms (possible, classic or probable by Katz hand diagram ( )) 6

Table 3

 3 Frequency and proportion of CTS using the selected case definitions in the PrediCTS study (n 1097)

	=

* PE Physical examination abnormality = ** NCS Nerve conduction study abnormality =

Table 4

 4 Concordance between the selected case definitions using the PrediCTS study data.

	Tested criteria	Reference criteria	True positive	False positive	False negative	True negative	Frequency (percent) of misclassification	Kappa
	( ) Franzblau-1 et al, 1993 7	Franzblau-2	75	46	0	976	46 (4.4)	0.74
		Matte	63	58	9	967	67 (6.5)	0.62
		Harrington	44	77	7	969	84 (8.3)	0.48
		Katz	35	86	6	970	92 (9.2)	0.40
		Sluiter	35	86	5	971	91 (9.0)	0.40
		Rempel	24	97	3	973	100 (10.0)	0.30

[START_REF] Descatha | Physical Examination is Minimally Useful in Defining Carpal Tunnel Syndrome in Population Based Studies[END_REF]

[START_REF] Descatha | Physical Examination is Minimally Useful in Defining Carpal Tunnel Syndrome in Population Based Studies[END_REF]

[START_REF] Descatha | Physical Examination is Minimally Useful in Defining Carpal Tunnel Syndrome in Population Based Studies[END_REF]

[START_REF] Descatha | Physical Examination is Minimally Useful in Defining Carpal Tunnel Syndrome in Population Based Studies[END_REF]

[START_REF] Descatha | Physical Examination is Minimally Useful in Defining Carpal Tunnel Syndrome in Population Based Studies[END_REF]

[START_REF] Descatha | Physical Examination is Minimally Useful in Defining Carpal Tunnel Syndrome in Population Based Studies[END_REF]

Ackowledgements:

This study was supported by research funding from the Centers for Disease Control/National Institute of Occupational Safety and Health (R01 OH008017-01) and from the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH), and NIH Roadmap for Medical Research (UL1 RR024992). Financial support for Justin Coomes was provided by the Doris Duke Charitable Foundation.


Mohammed Lemou

Florian Méhats

A boundary matching micro/macro decomposition for kinetic equations Une nouvelle décomposition micro/macro adaptée au bord pour les équations cinétiques

We introduce a new micro/macro decomposition of collisional kinetic equations which naturally incorporates the exact space boundary conditions. The idea is to write the distribution fonction f in all its domain as the sum of a Maxwellian adapted to the boundary (which is not the usual Maxwellian associated with f ) and a reminder kinetic part. This Maxwellian is defined such that its 'incoming' velocity moments coincide with the 'incoming' velocity moments of the distribution function. Important consequences of this strategy are the following. i) No artificial boundary condition is needed in the micro/macro models and the exact boundary condition on f is naturally transposed to the macro part of the model. ii) It provides a new class of the so-called 'Asymptotic preserving' (AP) numerical schemes: such schemes are consistent with the original kinetic equation for all fixed positive value of the Knudsen number ε, and if ε → 0 with fixed numerical parameters then these schemes degenerate into consistent numerical schemes for the various corresponding asymptotic fluid or diffusive models. Here, the strategy provides AP schemes not only inside the physical domain but also in the space boundary layers. We provide a numerical test in the case of a diffusion limit of the one-group transport equation, and show that our AP scheme recovers the boundary layer and a good approximation of the theoretical boundary value, which is usually computed from to the so-called Chandrasekhar function.

Résumé

Nous introduisons une nouvelle décomposition micro/macro pour les équations cinétiques collisionnelles qui tient compte de façon exacte des conditions aux bords en espace. L'idée est de décomposer la fonction de distribution f en une Maxwellienne adaptée au bord (qui n'est pas la Maxwellienne usuelle associée à f ) et une partie cinétique restante. Cette nouvelle Maxwellienne est définie de sorte que ses moments 'entrants' en vitesse soient les mêmes que les moments 'entrants' de f . Des conséquences importantes de cette stratégie sont les suivantes. i) Aucune condition artificielle n'est nécessaire et les conditions de bord sur f sont naturellement transposées sur la partie macroscopique de la décomposition. ii) La stratégie produit une nouvelle classe de schémas dits 'Asymptotic preserving' (AP) : ces schémas sont consistants avec le modèle cinétique original pour toute valeur positive du nombre de Knudsen ε, et si ε → 0 et les paramètres numériques sont fixés, alors ces schémas dégénèrent en des schémas qui sont consistants avec les divers modèles asymptotiques associés. Ici, les schémas obtenus reproduisent très bien les couches limites diffusives. Nous donnons un résultat numérique dans le cas d'une limite de diffusion et montrons que la valeur au bord théorique (calculée habituellement à partir de la fonction de Chandrasekhar) est bien approchée numériquement.

Version française abrégée

Le but de ce travail est d'introduire une nouvelle décomposition micro/macro pour les équations cinétiques, capable par construction d'intégrer les conditions aux bords exactes du modèle. Dans les approches usuelles, les modèle macrosopiques (asymptotiques ou aux moments) utilisent généralement des moyennes en vitesse de la fonction f ainsi que leurs flux, et les valeurs des ces moyennes au bord ne peuvent être en général déduites de la condition au bord sur la fonction distribution f . En effet, si la condition au bord en espace est de type Dirichlet sur f , alors la valeur de f au bord ne peut être imposée que pour les vitesses entrantes. Par conséquent les moyennes (sur tout le domaine des vitesses) de f ne sont pas données a priori, et des calculs supplémentaires (problème de Milne) et/ou des conditions artificielles sont en général nécessaires pour bien poser le modèle macroscopique.

La stratégie présentée ici permet de résoudre ce problème et constitue une base pour le développement des schémas dit 'Asymptotic Preserving' (AP) qui ont en plus la spécifité importante de pouvoir intégrer les conditions aux bords de façon exacte. Avant de détailler la stratégie dans les sections suivantes, nous rappelons brièvement la problématique des schémas multi-échelles dans ce contexte. Quand le nombre de Knudsen ε devient petit, une raideur apparaît dans ces modèles qui fait en sorte que leurs simulations numériques par des schémas explicites en temps deviennent rapidement inaccessibles (contraintes de type ∆t = O(ε) ou O(ε 2 )). Le problème est alors de construire des schémas numériques pour l'équation cinétique considérée qui s'affranchissenent de ces contraintes et dégénèrent en des discrétisations consistantes avec les modèles asymptotiques quand ε → 0. Plusieurs travaux ont été effectués à ce propos, voir par exemple [START_REF] Jin | Uniformly accurate diffusive relaxation schemes for multiscale transport equations[END_REF][START_REF] Klar | An asymptotic-induced scheme for nonstationary transport equations in the diffusion limit[END_REF][START_REF] Klar | An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit[END_REF] pour les limites de diffusion. Une bibliographie plus complète sera donnée dans une version détaillée de ce travail [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF]. Une approche générale a été introduite dans [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics[END_REF][START_REF] Lemou | A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit[END_REF][START_REF] Lemou | Relaxed micro/macro schemes for kinetic equations[END_REF], permettant de construire des schémas numériques qui sont consistants avec les modèles cinétiques étudiés pour toutes les valeurs de ε > 0, et dégénèrent quand ε → 0 en des discrétisations consistantes avec les systèmes d'Euler, de Navier-Stokes compressibles, ou avec la limite de diffusion. Elle est basée sur la décomposition micro-macro qui permet d'écrire de manière équivalente l'équation cinétique de départ sous la forme d'un système couplant une équation fluide ou de diffusion avec une équation sur la partie cinétique restante.

Le but ici est de compléter cette série de travaux sur les schémas micro/macro, en développant une nouvelle stratégie qui tient compte des conditions de bord en espace de façon exacte. Elle consiste à choisir une partie macrosopique 'intégrée sur les vitesses entrantes', qui est compatible avec les conditions aux bords, et de la coupler avec la partie cinétique restante. En plus d'intégrer naturellement les conditions aux bords, cette stratégie constitue une base pour la construction des schémas numériques qui ont toutes les bonnes propriétés des schémas micro/macro décrits ci-dessus. Cette note donne les grandes lignes de la stratégie qui sera développée en détail dans [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF]. Nous donnons aussi les résultats d'un premier test numérique dans le cas d'une limite diffusive, qui montre que le schéma construit est bien AP et permet de retrouver la bonne couche limite au bord. Des expériences numériques plus complètes seront effectuées dans [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF].

Boundary matching micro/macro formulations of kinetic equations: the general problem

It is a usual challenge to design efficient numerical schemes for kinetic equations which are consistent with the kinetic model for all positive value of the Knudsen number ε, and degenerate into consistent schemes with the asymptotic models (compressible Euler and Navier-Stokes, diffusion, etc) when ε → 0. On this subject, many works can be quoted, see for instance [START_REF] Jin | Uniformly accurate diffusive relaxation schemes for multiscale transport equations[END_REF][START_REF] Klar | An asymptotic-induced scheme for nonstationary transport equations in the diffusion limit[END_REF][START_REF] Klar | An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit[END_REF] and a more complete bibliography in the forthcoming paper [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF]. Recently [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics[END_REF][START_REF] Lemou | A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit[END_REF][START_REF] Lemou | Relaxed micro/macro schemes for kinetic equations[END_REF], a general strategy based on the micro/macro decomposition was introduced. It consists in writing the kinetic equation as a system coupling a macroscopic part (say a Maxwellian) and a reminder kinetic part. The main advantage of this method is its robustness and easy adaptability. However, the general problem of transfering boundary conditions from the distribution function f to the two parts of the micro/macro systems has not been solved and artificial boundary conditions were needed.

Our aim is to develop a new micro/macro decomposition of collisional kinetic equations which naturally incorporates the exact space boundary conditions (BC). In fact, it is well known that during the derivation of a macroscopic model from kinetic equation (via moments or Chapman-Enskog approach), the exact boundary conditions on f are generally lost. Indeed, if one takes moments of the kinetic equation, then the values of the fluxes at the boundary are required. But, in general (for inflow BC for instance), the distribution function f is known on the boundary for only incoming velocities and, therefore, their moments are not prescribed at the boundary.

The idea is to decompose the distribution fonction f in its domain as the sum of a Maxwellian part adapted to the boundary (which is not the usual Maxwellian associated with f ) and a reminder kinetic part. This Maxwellian is defined on the whole domain such that its 'incoming' velocity moments coincide with the 'incoming' velocity moments of the distribution function. Important consequences of this strategy are the following. i) No artificial boundary condition is needed in the micro/macro models and the exact boundary conditions on f are naturally shared by the macro part and the kinetic part. Note that the traditional approaches cannot provide exact boundary conditions for the asymptotic and kinetic models separately, and extra calculations (Milne problems for instance) or artificial boundary conditions are necessary in general. ii) It provides a new class of the so-called 'Asymptotic preserving' (AP) numerical schemes: such schemes are consistent with the original kinetic equation for all fixed positive value of the Knudsen number ε, and when ε → 0 with fixed numerical parameters, these schemes degenerate into consistent numerical schemes for the various corresponding asymptotic models. A further fundamental property of the strategy developed here is that it provides AP schemes not only inside the physical domain but also in the space boundary layers with exact boundary conditions. We shall refer to this property as 'Boundary Asymptotic Preserving' (BAP). At the end of this Note, we provide a numerical test which illustrates this property. We emphasize that our scheme does not use any artificial boundary condition: it provides at the diffusion limit a good approximation of the theoretical boundary value derived from the Chandrasekhar function, without injecting this value in our scheme. This Note summarizes the main lines of the strategy and announces a more complete presentation [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF], where numerical discretizations of this strategy will be constructed and implemented with a large variety of numerical tests.

The case of the diffusion scaling

Let Ω be a domain in the position (physical) space R d , with boundary ∂Ω, and V be a domain in the velocity space R d , endowed with a measure dµ. We consider a function ω : Ω × V → R such that for all x ∈ ∂Ω and v ∈ V , ω(x, v) has the same sign as n(x) • v, where n(x) is the outgoing normal vector to ∂Ω at x. Let us give explicit examples of ω(x, v) for specific geometries. For a ball centered at the origin, one can take ω(x, v) = x • v. For a half plane x 1 > 0, one can choose ω(x, v) = (-v 1 , 0, ..., 0). In dimension one, for the interval (0, 1), one can take ω(x, v) = (2x -1)v. We then consider the following transport equation in a diffusive scaling

ε∂ t f + v • ∇ x f = 1 ε Lf, t > 0, (x, v) ∈ Ω × V, f | t=0 = f init (1)
where f is the distribution function of the particles that depends on time t > 0, on position x ∈ Ω, and on velocity v ∈ V . The linear operator L acts on the velocity dependence of f and describes the interactions of particles with the medium. We assume that there exists a positive equilibrium function E = E(v) with E := V Edµ = 1 and satisfying vE = 0, and that the collision operator L is non-positive and selfadjoint in L 2 (V, E -1 dµ); with null space and image given by N (L) = Span{E} = {f = ρE, where ρ := f := V f dµ}, R(L) = (N (L)) ⊥ = {f such that f = 0}. When ε goes to 0 in (1), it is easily seen that f converges to an equilibrium state f 0 = ρ 0 (t, x)E(v). The diffusion limit is the equation satisfied by the density ρ 0 and is classically given by

∂ t ρ 0 + ∇ x • (κ∇ x ρ 0 ) = 0, with κ = vL -1 (vE) . (2)
Following [START_REF] Lemou | A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit[END_REF] and [START_REF] Lemou | Relaxed micro/macro schemes for kinetic equations[END_REF], equation ( 1) can be written in a micro-macro equivalent form via the decomposition f = ρE + g, with ρ(t, x) = f and g = f -ρE (g is not necessarily small as in [START_REF] Lemou | A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit[END_REF]). Let Π be the orthogonal projector in L 2 (E -1 dµ) onto the nullspace of L: Πφ = φ E. Then, inserting this decomposition into the kinetic equation and applying Π and I -Π successively, one gets after direct computations

∂ t ρ + 1 ε ∇ x • vg = 0, ∂ t g + 1 ε (I -Π)(v • ∇ x g) = 1 ε 2 Lg -εEv • ∇ x ρ) . (3)
We now emphasize that in this formulation, the space boundary condition on ρ and g are not known because they cannot be inferred from the boundary conditions on f in general. Indeed, in the typical case of incoming boundary conditions, we impose

f (t, x, v) = f b (t, x, v), ∀t > 0, ∀(x, v) ∈ ∂Ω × V such that ω(x, v) < 0. (4)
It is therefore clear that the values of ρ(t, x) = V f (t, x, v)dµ cannot be determined a priori on the boundary ∂Ω, since f is only known for incoming velocities v, i.e. such that ω(x, v) < 0. Our aim in this work is to develop a new micro/macro decomposition which is able to incorporate the boundary conditions in an exact way. To this purpose, let us introduce some few further notations:

V -(x) = {v ∈ V, ω(x, v) < 0}, V + (x) = V \V -(x). ( 5
)
The idea is now the following. Instead of looking for an equation on ρ as usual, we seek an equation on the following 'boundary matching' density and perform the corresponding micro/macro decomposition:

ρ(t, x) = f (t, x, •) V- E(t, x, .) V- , • V-= V- • dµ, f = ρE + g. (6)
When ε → 0, we know that the solution f of ( 1) is (at least formally) close to ρE except in initial or boundary layers. Therefore the 'boundary matching' density ρ will be close to ρ for small ε. This shows that ( 6) still a decomposition of f into an asymptotic part (macro part) and a kinetic part (micro part).

In order to derive the system of equations satisfied by ρ and g from ( 1), we first integrate (1) on V -and get

∂ t ρ + 1 ε vE V-• ∇ x ρ + 1 ε v • ∇ x g V-= 1 ε 2 Lg V-. (7)
Substracting this from the equation on f , we obtain the equation on g:

∂ t g + 1 ε v • ∇ x g -v • ∇ x g V-+ 1 ε vE -vE V-• ∇ x ρ = 1 ε 2 Lg -Lg V-. (8)
System ( 7)-( 8) can be replaced by a the more convenient (and still equivalent) system in terms of ρ = f = V f dµ and g = f -ρE as follows:

∂ t ρ + 1 ε v • ∇ x g = 0, ∂ t g + 1 ε v • ∇ x g -v • ∇ x g V-+ 1 ε vE -vE V-• ∇ x ρ = 1 ε 2 Lg -Lg V-. (9)
Note that ρ is linked to ρ and g by the relations ρ = ρ -g = ρ -g V+ and f = ρ -g V+ + g.

One main interest of this new micro/macro formulation (9) for the original kinetic equation ( 1) is the following: the fluxes involved in (9) only concern the quantities g or ρ, and not ρ. This means that numerical schemes of this formulation would only need the values of ρ and g at the space boundary which are completely known from the original boundary condition (4) on f .

The case of a fluid scaling

We now consider non linear kinetic equations with a fluid scaling

∂ t f + v • ∇ x f = 1 ε Q(f, f ), t > 0, (x, v) ∈ Ω × V, f (t = 0, x, v) = f init (x, v), (10)
where Ω ⊂ R d and V = R d . The collision operator Q is a quadratic operator acting only on the velocity dependence of the distribution function f . Fundamental examples are the well known Boltzmann kernel for rarefied gases and the Fokker-Planck-Landau operators for plasmas. In all what follows, we use the notations

m(v) = (1, v, |v| 2 /2) T , and g = R d g(v) dv (11)
for any scalar or vector function g = g(v). It is well known that the Boltzmann and Landau operators Q(f, f ) have important physical properties of conservation and entropy: for all f ≥ 0, we have mQ(f, f ) = 0 and Q(f, f )log(f ) ≤ 0. It is also well known that the equilibrium functions (f such that Q(f, f ) = 0) are Maxwellians:

M (U )(v) = ρ(2πT ) -d/2 exp -|v -u| 2 /2T , U = (ρ, ρu, ρ|u| 2 /2 + (d/2)ρT ) = mM (U ) . ( 12
)
To any distribution function f , we shall associate its Maxwellian M = M [f ] = M (U ), that is the fonction of the form (12) which has the same moments as f :

U = mf = mM [f ] = mM (U ) .
When ε goes to 0, f approaches a Maxwellian. At the first order in ε, the solution f approaches that of the compressible Euler system of gas dynamics. This is formally obtained by integrating the kinetic equation (10) against m(v) and replacing f by its first order approximation: the Maxwellian which has the same first moments as f . At the second order in ε, a standard Chapman-Enskog expansion leads to the compressible Navier-Stokes system. In order to design numerical schemes which are able to reproduce these asymptotics, a general class of numerical schemes has been introduced in [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics[END_REF] and [START_REF] Lemou | Relaxed micro/macro schemes for kinetic equations[END_REF] on the basis of micro/macro decompositions. The strategy in [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics[END_REF][START_REF] Lemou | Relaxed micro/macro schemes for kinetic equations[END_REF] consists in decomposing f as f = M (U ) + g and transform (10) into an equivalent system of equations on M and g. Then, discretizations of this formulations lead to a class of AP numerical schemes which are consistent with the fluid limit in the asymptotics ε → 0. However, similarly as above for the diffusion limit, the exact space boundary conditions are lost in this decomposition, and one should compute approximate BC for the asymptotic models. To solve this problem, we introduce here a more suited decomposition matching the boundary. We first define the new moments U by U (t, x) = m(v)f (t, x, v) V-where V -is defined in [START_REF] Klar | An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit[END_REF], and the corresponding equilibrium M is the (uniquely defined, see [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF] for details) Maxwellian such that V-m(v)M dv = U (t, x). In some sense, this strategy has some similarities with the half-moment method developed in [START_REF] Dubroca | Prise en compte d'un fort déséquilibre cinétique par un modèle aux demi-moments [Half-moment model taking into account strong kinetic non-equilibrium[END_REF]. Note that M is of the form (12), i.e. M = M ( U ), but in general U is different from U . In fact, since the domain of integration is now V -and not R d , the relation between the parameters of M and U is not explicit in general. We now introduce the following micro/macro decomposition:

f = M + g, V- m(v)M dv = U (t, x) = V- m(v)f (t, x, v)dv. (13)
When ε → 0, it is known (at least formally) that the solution f of (10) approaches its classical Maxwellian M : f -M = O(ε). This means that V-m(v)(f -M ) = O(ε), and then from (13), we have V-m(v)(M -M ) = O(ε). This gives d + 2 relations between the two Maxwellians M and M , each of these Mawellians being completely determined by d + 2 parameters (from ( 12)). Therefore M -M = O(ε). In some sense, this justifies the name 'micro/macro' for this new decomposition.

In order to transform (10) into a coupled system on M and g, we first introduce a definition. For each Maxwellian M , we define Π as the orthogonal projection on Span{M , vM ,

|v| 2 M } in L 2 M -1 1 V-. This
projector has an explicit expression, see [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF]. We now insert the decomposition (13) into (10):

∂ t M + v • ∇ x M + ∂ t g + v • ∇ x g = 1 ε L M g + Q(g, g) , L M g = 2Q(M , g).
Now instead of writing a system on M and g, it turns out to be more convenient to rather write a system on the full moments U = mf and g. To obtain it, we simply integrate (10) against m(v) on the whole velocity domain V = R d for the equation on U , and apply I -Π to (10) to get the equation on g:

∂ t U + ∇ x • R d vm(v)M dv + ∇ x • R d vm(v)gdv = 0, ∂ t g + I -Π (v • ∇ x M ) + I -Π (v • ∇ x g) = 1 ε I -Π L M g + Q(g, g) . (14)

A numerical test

In this Note, we propose to validate the strategy in the case of the diffusion limit only. A more detailed numerical validation will be done in [START_REF] Lemou | Asymptotic Preserving schemes for kinetic equations including boundary layers[END_REF]. We consider the simple framework of one-dimensional spaces x ∈ Ω = [0, 1] and v ∈ V = [-1, 1] with dµ = 1 2 dv, and we take Lf = 1 2 1 -1 f dv -f = ρ -f . The initial data is f init (x, v) = 0 and the boundary data are f (t, 0, v) = v for v > 0 and f (t, 1, v) = 0 for v < 0. On Figure 1, we plot the densities ρ obtained by the AP micro-macro scheme matching the boundary, for the values ε ∈ {1, 0.5, 0.2, 0.05, 0.0001} and compare them with the reference solutions. For ε ∈ {1, 0.5, 0.2, 0.05}, the reference solutions are computed with a highly resolved explicit scheme for [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics[END_REF]. For ε = 0.0001, the reference solution is computed by the diffusion equation ( 2) with the theoretical value of the Dirichlet boundary condition, which is ρ(0) = 0.71043... at x = 0, calculated thanks to the usual Chandrasekhar function, and ρ(1) = 0 at x = 1. The numerical results show that our scheme is AP in the diffusion regime and is able to reproduce efficiently the boundary layer. The present numerical experiment confirmes that the scheme is 'BAP' in the sense described above: without introducing any artificial boundary condition, we recover both the diffusive regime and the boundary layer. Left: numerical solutions in the kinetic and transition regimes, ε = 1, ε = 0.5, ε = 0.2. The density for the micro/macro scheme coincides with the reference calculated with a highly resolved explicit scheme. Right: numerical solutions in the transition and diffusion regimes. The density for the micro/macro scheme coincides with the reference. For ε = 0.05, the reference is calculated with the highly resolved explicit scheme. For ε = 0.0001, the reference is calculated by the diffusion equation with the Chandrasekhar value at the boundary. Both figures are plotted at time t = 0.4 and all micro/macro schemes are used with a uniform grid of 100 points in x.

  Note that, since f = M + g, the parameters (ρ, u, T ) of the Maxwellian M (according to definition (12)) are linked to U and g by the simple relation:(ρ, ρu, ρ|u| 2 /2 + (d/2)ρT ) = U -m(v)g .On the other hand, this relation relation cannot determine M at the boundary because U is not known at the boundary. However, by construction, M is completely computed at the boundary from the relation m(v)M V-= m(v)f b V-, where f b is the incoming boundary condition (4).

Figure 1 .

 1 Figure1. Left: numerical solutions in the kinetic and transition regimes, ε = 1, ε = 0.5, ε = 0.2. The density for the micro/macro scheme coincides with the reference calculated with a highly resolved explicit scheme. Right: numerical solutions in the transition and diffusion regimes. The density for the micro/macro scheme coincides with the reference. For ε = 0.05, the reference is calculated with the highly resolved explicit scheme. For ε = 0.0001, the reference is calculated by the diffusion equation with the Chandrasekhar value at the boundary. Both figures are plotted at time t = 0.4 and all micro/macro schemes are used with a uniform grid of 100 points in x.

Email addresses: mohammed.lemou@univ-rennes1.fr (Mohammed Lemou), florian.mehats@univ-rennes1.fr (Florian Méhats).

Acknowledgement. The authors were supported by the french ANR project CBDif. M. Lemou acknowledges support from the project 'Défis émergents' funded by the university of Rennes 1. F. Méhats acknowledges support from the french ANR project QUATRAIN and from the INRIA project IPSO.